Is it true that All equations have only one solution.?

Simple quadratic equations can have zero, one, or two solutions (e.g., x² = 16 has two solutions: x = 4 and x = -4).

Is it true that The hypotenuse is always the longest side in any triangle.?

The hypotenuse is specifically the longest side in a *right-angled* triangle, opposite the right angle.

Year 8 Mathematics: Equations, Pythagoras, Rates, Ratios & Financial Math

Mastering fundamental mathematical concepts including algebraic equations, geometric theorems, proportional reasoning, and financial calculations is crucial for Year 8 academic success.

Core Principles

Equations: Maintain balance by performing the same operation on both sides to isolate the unknown.
Pythagoras' Theorem: In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a² + b² = c²).
Rates and Ratios: Compare quantities with the same or different units using simplified forms and unitary methods.
Financial Mathematics: Understand percentages, mark-ups, discounts, and profit/loss calculations for managing money.

Action Steps

To solve equations, perform inverse operations on both sides to isolate the variable (e.g., add to undo subtraction, divide to undo multiplication).
To find the hypotenuse using Pythagoras' Theorem, square the lengths of the two shorter sides, add them, and then take the square root of the sum.
To find a shorter side using Pythagoras' Theorem, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
To simplify ratios, divide all parts by their highest common factor or convert to a common unit first.
To solve problems involving rates, use the unitary method to find the value of one unit (e.g., cost per kg, speed per hour) and then scale up or down.
To calculate percentage increase, find the percentage of the original value and add it to the original value.
To calculate percentage decrease, find the percentage of the original value and subtract it from the original value.

Formulas

Pythagoras' Theorem: $a^2 + b^2 = c^2$
Perimeter of a rectangle: $P = 2l + 2b$
Speed, distance, time: $d = s \times t$

Key Terms

Equation: A mathematical statement that asserts the equality of two expressions, typically containing variables.
Inequality: A mathematical statement comparing two expressions using symbols like >, <, ≥, or ≤.
Pythagorean Triple: A set of three positive integers (a, b, c) that satisfy the equation a² + b² = c².
Ratio: A comparison of two or more quantities, often expressed in simplest form or as a relationship between parts.
Rate: A ratio that compares two quantities measured in different units (e.g., speed, price per kg).
Unitary Method: A method for solving problems by first finding the value of a single unit.
Percentage: A fraction out of 100, represented by the symbol %.

Pro Tips

When solving equations with negative coefficients, multiply both sides by -1 to make the coefficient positive, remembering to reverse the inequality sign if applicable.
For problems involving scale, ensure units are consistent before calculating; convert map distances to real-world units or vice versa.
When dealing with percentages, remember that 'percentage' means 'out of 100', so convert percentages to fractions or decimals for calculations.

Pitfalls to Avoid

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number causes incorrect solutions for inequalities.
Incorrectly applying Pythagoras' Theorem by adding the squares of the two shorter sides when trying to find a shorter side, instead of subtracting.
Calculating percentages without converting to a common unit first, leading to inaccurate comparisons or results.

Myth vs Reality

All equations have only one solution.: Simple quadratic equations can have zero, one, or two solutions (e.g., x² = 16 has two solutions: x = 4 and x = -4).
The hypotenuse is always the longest side in any triangle.: The hypotenuse is specifically the longest side in a *right-angled* triangle, opposite the right angle.

Real World Examples

An orange costs 5c more than an apple, and together they cost $1.15.: This problem is solved using equations to find the individual cost of an apple (x = $0.55) and an orange (x + 5 = $0.60).
A map has a scale of 1:20000, and a distance on the map is 5 mm.: To find the actual distance, multiply the map distance by the scale factor: 5 mm × 20000 = 100000 mm = 100 m.
A pair of shoes is discounted by 20%, and the sale price is $120.: This requires using the unitary method to find the original price: If $120 represents 80% (100% - 20%), then 1% is $1.50, and 100% is $150.

Statistics

Cost of an apple in a problem-solving example: 55 cents
Cost of an orange in a problem-solving example: 60 cents
Weight of 6 identical books: 1.2 kg